abstract class RomanDigit 

case object I extends RomanDigit 

case object V extends RomanDigit 

case object X extends RomanDigit 

case object L extends RomanDigit 

case object C extends RomanDigit 

case object D extends RomanDigit 

case object M extends RomanDigit 

type RomanNumeral = List[RomanDigit] 

def optionlist[A](xs: List[Option[A]]): Option[List[A]] = xs match {

  case Nil => Some(Nil)

  case x::xs => (x, optionlist(xs)) match {

    case (None, _) => None

    case (_, None) => None

    case (Some(y), Some(ys)) => Some(y::ys)  

  }

}

def char2romandigit(c: Char): Option[RomanDigit] = c match {

  case 'I' => Some(I)

  case 'V' => Some(V)

  case 'X' => Some(X)

  case 'L' => Some(L)

  case 'C' => Some(C)

  case 'D' => Some(D)

  case 'M' => Some(M)

  case _ => None

}

def String2RomanNumeral(s: String) : Option[RomanNumeral] = 

  optionlist(s.toList.map(char2romandigit))

String2RomanNumeral("IIII")

String2RomanNumeral("IV")

String2RomanNumeral("VI")

String2RomanNumeral("IX")

String2RomanNumeral("MCMLXXIX")

String2RomanNumeral("MCMXLIV") 

String2RomanNumeral("M C M X L I V")  // None

////////////////

def RomanNumeral2Int(rs: RomanNumeral): Int = rs match { 

  case Nil => 0

  case M::r    => 1000 + RomanNumeral2Int(r)  

  case C::M::r => 900 + RomanNumeral2Int(r)

  case D::r    => 500 + RomanNumeral2Int(r)

  case C::D::r => 400 + RomanNumeral2Int(r)

  case C::r    => 100 + RomanNumeral2Int(r)

  case X::C::r => 90 + RomanNumeral2Int(r)

  case L::r    => 50 + RomanNumeral2Int(r)

  case X::L::r => 40 + RomanNumeral2Int(r)

  case X::r    => 10 + RomanNumeral2Int(r)

  case I::X::r => 9 + RomanNumeral2Int(r)

  case V::r    => 5 + RomanNumeral2Int(r)

  case I::V::r => 4 + RomanNumeral2Int(r)

  case I::r    => 1 + RomanNumeral2Int(r)

}

RomanNumeral2Int(List(I,I,I,I))         // 4

RomanNumeral2Int(List(I,V))             // 4

RomanNumeral2Int(List(V,I))             // 6

RomanNumeral2Int(List(I,X))             // 9

RomanNumeral2Int(List(M,C,M,L,X,X,I,X)) // 1979

RomanNumeral2Int(List(M,C,M,X,L,I,V))   // 1944

def String2Int(s: String): Option[Int] = 

  String2RomanNumeral(s).map(RomanNumeral2Int(_))

String2Int("IIII")      // 4 invalid 

String2Int("IV")        // 4

String2Int("VI")        // 6

String2Int("IX")        // 9

String2Int("MCMLXXIX")  // 1979

String2Int("MCMXLIV")   // 1944

String2Int("")          // 0

String2Int("MDCLXI")    // 1661

String2Int("MMMCMXCIX") // 3999

String2Int("XLVIII")    // 48

String2Int("MMVIII")    // 2008

String2Int("MMXI")      // 2011 

String2Int("MIM")       // 1999

String2Int("MCMLVI")    // 1956 

String2Int("XXCIII")    // ?? 103 / 83

String2Int("LXXIIX")    // ?? 80 / 78

String2Int("IIIIX")     // ?? 12 invalid  

String2Int("IIXX")      // ?? 20 / 18 invalid

String2Int("III") 	// 3

String2Int("XXX") 	// 30

String2Int("CCC") 	// 300

String2Int("MMM") 	// 3000

String2Int("VII") 	// 7

String2Int("LXVI") 	// 66

String2Int("CL") 	// 150

String2Int("MCC") 	// 1200

String2Int("IV") 	// 4

String2Int("IX") 	// 9

String2Int("XC") 	// 90

String2Int("ICM") 	// ?? 901

String2Int("CIM") 	// ?? 899

String2Int("MDCLXVI")	// 1666

String2Int("VIV")       //9, but should be written as IX 

String2Int("IVX")       // 14 invalid

// error cases

String2Int("MC?I")

String2Int("abc")

def runsAllowed(r: RomanDigit) = r match {

  case I | X | C | M => true

  case V | L | D => false

}

def norunsAllowed(r: RomanDigit) = !runsAllowed(r)

def isValidNumeral(digitList: RomanNumeral): Bool = digitList match {

    // empty list is valid

  case Nil => true

    // a following digit that is equal or larger is an error

  case d1::d2::_ if (d1 <= d2)  => false 

    // A single digit is always allowed

  case _::ds => isValidNumeral(ds) 

}